Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.
(3u) • v

Verified step by step guidance

1. Recall that the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by the formula: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \]
2. First, calculate the vector \( 3\mathbf{u} \) by multiplying each component of \( \mathbf{u} = \langle -2, 1 \rangle \) by 3: \[ 3\mathbf{u} = \langle 3 \times (-2), 3 \times 1 \rangle = \langle -6, 3 \rangle \]
3. Now, use the dot product formula to find \( (3\mathbf{u}) \cdot \mathbf{v} \), where \( \mathbf{v} = \langle 3, 4 \rangle \): \[ (3\mathbf{u}) \cdot \mathbf{v} = (-6)(3) + (3)(4) \]
4. Simplify the expression by performing the multiplications inside the dot product: \[ (-6)(3) + (3)(4) = -18 + 12 \]
5. Finally, add the results to get the value of the dot product: \[ -18 + 12 \]

Key Concepts

Vector Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For example, multiplying vector u = 〈-2, 1〉 by 3 results in 〈-6, 3〉. This operation scales the vector's magnitude without changing its direction.

Dot Product of Vectors

The dot product of two vectors is the sum of the products of their corresponding components. For vectors a = 〈a1, a2〉 and b = 〈b1, b2〉, the dot product is a1*b1 + a2*b2. It results in a scalar and measures the extent to which the vectors point in the same direction.

Properties of Dot Product with Scalar Multiplication

The dot product is distributive over scalar multiplication, meaning (c*u) • v = c*(u • v), where c is a scalar. This property allows simplification by factoring out scalars before computing the dot product, making calculations more efficient.